Exactly solving some typical Riemann–Liouville fractional models by a general method of separation of variables

Abstract

Finding exact solutions for Riemann–Liouville (RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.

Optical solitons of the (2+1)-dimensional nonlinear Schrödinger equation with spatio-temporal dispersion in quadratic-cubic media

“…Whereas, not all forms of the exact solutions are acquired. Thereupon we use the method of complete discrimination system for polynomial proposed by Liu [39,[41][42][43][44] to solve equation (3) and obtain some new exact solutions, which are trigonometric function solutions, solitary wave solutions and elliptic functions double periodic solutions. In the meantime, the topological stability of all solutions is analyzed.…”

Optical solitons of Sasa-Satsuma model in birefringent fibers

“…[30,31], we can get new solutions, they are singular rational patterns and elliptic function patterns. This method is proposed by Liu [32][33][34][35][36][37][38][39] and are widely used to solve exact solutions of mathematical physics equations [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. From the new patterns we obtained, we can better understand the dynamic behaviors of the model.…”